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The focus on problem solvingand inquiry in the learning activities also provides opportunities for students to: • find enjoyment in mathematics; • develop confidence in learning and usin

Number Sense and Numeration,

Grades 4 to 6

Volume 4 Division

A Guide to Effective Instruction

in Mathematics, Kindergarten to Grade 6

2006

Every effort has been made in this publication to identify mathematics resources and tools(e.g., manipulatives) in generic terms In cases where a particular product is used by teachers

in schools across Ontario, that product is identified by its trade name, in the interests of clarity.Reference to particular products in no way implies an endorsement of those products bythe Ministry of Education

Number Sense and Numeration,

Relating Mathematics Topics to the Big Ideas 6

The Mathematical Processes 6

Addressing the Needs of Junior Learners 8

Learning About Division in the Junior Grades 11 Introduction 11

Interpreting Division Situations 13

Relating Multiplication and Division 14

Using Models to Represent Division 14

Learning Basic Division Facts 16

Considering the Meaning of Remainders 16

Developing a Variety of Computational Strategies 17

Developing Estimation Strategies for Division 23

A Summary of General Instructional Strategies 24

Appendix 4–1: Using Mathematical Models to Represent Division 25 References 31 Learning Activities for Division 33 Introduction 33

Grade 4 Learning Activity 35

Grade 5 Learning Activity 48

Grade 6 Learning Activity 58

Number Sense and Numeration, Grades 4 to 6 is a practical guide, in six volumes, that teachers

will find useful in helping students to achieve the curriculum expectations outlined for Grades

4 to 6 in the Number Sense and Numeration strand of The Ontario Curriculum, Grades 1–8:

Mathematics, 2005 This guide provides teachers with practical applications of the principles

and theories behind good instruction that are elaborated on in A Guide to Effective Instruction

in Mathematics, Kindergarten to Grade 6, 2006

The guide comprises the following volumes:

• Volume 1: The Big Ideas

• Volume 2: Addition and Subtraction

• sample learning activities dealing with division for Grades 4, 5, and 6

A glossary that provides definitions of mathematical and pedagogical terms used throughoutthe six volumes of the guide is included in Volume 1: The Big Ideas Each volume also contains

a comprehensive list of references for the guide

The content of all six volumes of the guide is supported by “eLearning modules” that areavailable at www.eworkshop.on.ca The instructional activities in the eLearning modulesthat relate to particular topics covered in this guide are identified at the end of each of thelearning activities (see pp 44, 55, and 68)

Relating Mathematics Topics to the Big Ideas

The development of mathematical knowledge is a gradual process A continuous, cohesive program throughout the grades is necessary to help students develop an understanding of the “big ideas” of mathematics – that is, the interrelated concepts that form a framework for learning mathematics in a coherent way.

(The Ontario Curriculum, Grades 1–8: Mathematics, 2005, p 4)

In planning mathematics instruction, teachers generally develop learning opportunities related

to curriculum topics, such as fractions and division It is also important that teachers designlearning opportunities to help students understand the big ideas that underlie importantmathematical concepts The big ideas in Number Sense and Numeration for Grades 4 to 6 are:

• quantity • representation

• operational sense • proportional reasoning

• relationships Each of the big ideas is discussed in detail in Volume 1 of this guide

When instruction focuses on big ideas, students make connections within and between topics,and learn that mathematics is an integrated whole, rather than a compilation of unrelatedtopics For example, in a lesson about division, students can learn about the relationshipbetween multiplication and division, thereby deepening their understanding of the big idea

of operational sense

The learning activities in this guide do not address all topics in the Number Sense andNumeration strand, nor do they deal with all concepts and skills outlined in the curriculumexpectations for Grades 4 to 6 They do, however, provide models of learning activities thatfocus on important curriculum topics and that foster understanding of the big ideas in NumberSense and Numeration Teachers can use these models in developing other learning activities

The Mathematical Processes

The Ontario Curriculum, Grades 1–8: Mathematics, 2005 identifies seven mathematical processes

through which students acquire and apply mathematical knowledge and skills The matical processes that support effective learning in mathematics are as follows:

mathe-• problem solving • connecting

• reasoning and proving • representing

• reflecting • communicating

• selecting tools and computational strategiesThe learning activities described in this guide demonstrate how the mathematical processes

The learning activities also demonstrate that the mathematical processes are interconnected –for example, problem-solving tasks encourage students to represent mathematical ideas, toselect appropriate tools and strategies, to communicate and reflect on strategies and solutions,and to make connections between mathematical concepts

Problem Solving: Each of the learning activities is structured around a problem or inquiry.

As students solve problems or conduct investigations, they make connections between newmathematical concepts and ideas that they already understand The focus on problem solvingand inquiry in the learning activities also provides opportunities for students to:

• find enjoyment in mathematics;

• develop confidence in learning and using mathematics;

• work collaboratively and talk about mathematics;

• communicate ideas and strategies;

• reason and use critical thinking skills;

• develop processes for solving problems;

• develop a repertoire of problem-solving strategies;

• connect mathematical knowledge and skills with situations outside the classroom

Reasoning and Proving: The learning activities described in this guide provide opportunities

for students to reason mathematically as they explore new concepts, develop ideas, makemathematical conjectures, and justify results The learning activities include questions teacherscan use to encourage students to explain and justify their mathematical thinking, and toconsider and evaluate the ideas proposed by others

Reflecting: Throughout the learning activities, students are asked to think about, reflect on,

and monitor their own thought processes For example, questions posed by the teacherencourage students to think about the strategies they use to solve problems and to examinemathematical ideas that they are learning In the Reflecting and Connecting part of eachlearning activity, students have an opportunity to discuss, reflect on, and evaluate theirproblem-solving strategies, solutions, and mathematical insights

Selecting Tools and Computational Strategies: Mathematical tools, such as manipulatives,

pictorial models, and computational strategies, allow students to represent and do mathematics

The learning activities in this guide provide opportunities for students to select tools (concrete,pictorial, and symbolic) that are personally meaningful, thereby allowing individual students

to solve problems and represent and communicate mathematical ideas at their own level ofunderstanding

Connecting: The learning activities are designed to allow students of all ability levels to connect

new mathematical ideas to what they already understand The learning activity descriptionsprovide guidance to teachers on ways to help students make connections among concrete,pictorial, and symbolic mathematical representations Advice on helping students connect

procedural knowledge and conceptual understanding is also provided The problem-solvingexperiences in many of the learning activities allow students to connect mathematics to real-lifesituations and meaningful contexts

Representing: The learning activities provide opportunities for students to represent

mathe-matical ideas using concrete materials, pictures, diagrams, numbers, words, and symbols.Representing ideas in a variety of ways helps students to model and interpret problem situations,understand mathematical concepts, clarify and communicate their thinking, and make connec-tions between related mathematical ideas Students’ own concrete and pictorial representations

of mathematical ideas provide teachers with valuable assessment information about studentunderstanding that cannot be assessed effectively using paper-and-pencil tests

Communicating: Communication of mathematical ideas is an essential process in learning

mathematics Throughout the learning activities, students have opportunities to express matical ideas and understandings orally, visually, and in writing Often, students are asked

mathe-to work in pairs or in small groups, thereby providing learning situations in which studentstalk about the mathematics that they are doing, share mathematical ideas, and ask clarifyingquestions of their classmates These oral experiences help students to organize their thinkingbefore they are asked to communicate their ideas in written form

Addressing the Needs of Junior Learners

Every day, teachers make many decisions about instruction in their classrooms To makeinformed decisions about teaching mathematics, teachers need to have an understanding ofthe big ideas in mathematics, the mathematical concepts and skills outlined in the curriculumdocument, effective instructional approaches, and the characteristics and needs of learners.The following table outlines general characteristics of junior learners, and describes some of theimplications of these characteristics for teaching mathematics to students in Grades 4, 5, and 6

Characteristics of Junior Learners and Implications for Instruction

Area of Development Characteristics of Junior Learners Implications for Teaching Mathematics

Intellectualdevelopment

Generally, students in the junior grades:

• prefer active learning experiences thatallow them to interact with their peers;

• are curious about the world around them;

• are at a concrete operational stage ofdevelopment, and are often not ready tothink abstractly;

• enjoy and understand the subtleties

of humour

The mathematics program should provide:

• learning experiences that allow students

to actively explore and construct mathematical ideas;

• learning situations that involve the use

of concrete materials;

• opportunities for students to see thatmathematics is practical and important

in their daily lives;

• enjoyable activities that stimulate curiosityand interest;

• tasks that challenge students to reasonand think deeply about mathematical ideas

Physicaldevelopment

Generally, students in the junior grades:

• experience a growth spurt before

puber-ty (usually at age 9–10 for girls,

at age 10–11 for boys);

• are concerned about body image;

• are active and energetic;

• display wide variations in physical opment and maturity

devel-The mathematics program should provide:

• opportunities for physical movement andhands-on learning;

• a classroom that is safe and physicallyappealing

Psychologicaldevelopment

Generally, students in the junior grades:

• are less reliant on praise but stillrespond well to positive feedback;

• accept greater responsibility for theiractions and work;

• are influenced by their peer groups

The mathematics program should provide:

• ongoing feedback on students’ learningand progress;

• an environment in which students cantake risks without fear of ridicule;

• opportunities for students to acceptresponsibility for their work;

• a classroom climate that supports diversityand encourages all members to workcooperatively

Social development

Generally, students in the junior grades:

• are less egocentric, yet require individualattention;

• can be volatile and changeable in regard

to friendship, yet want to be part of asocial group;

• can be talkative;

• are more tentative and unsure of themselves;

• mature socially at different rates

The mathematics program should provide:

• opportunities to work with others in avariety of groupings (pairs, small groups,large group);

• opportunities to discuss mathematicalideas;

• clear expectations of what is acceptablesocial behaviour;

• learning activities that involve all studentsregardless of ability

(continued)

(Adapted, with permission, from Making Math Happen in the Junior Grades

Elementary Teachers’ Federation of Ontario, 2004.)

Characteristics of Junior Learners and Implications for Instruction

Area of Development Characteristics of Junior Learners Implications for Teaching Mathematics

Moraland ethicaldevelopment

Generally, students in the junior grades:

• develop a strong sense of justice andfairness;

• experiment with challenging the normand ask “why” questions;

• begin to consider others’ points of view

The mathematics program should provide:

• learning experiences that provide table opportunities for participation byall students;

equi-• an environment in which all ideas arevalued;

• opportunities for students to sharetheir own ideas and evaluate the ideas

of others

LEARNING ABOUT DIVISION IN THE JUNIOR GRADES

Introduction

Students’ understanding of division concepts andstrategies is developed through meaningful and purposeful problem-solving activities Solving a variety

of division problems and discussing various strategiesand methods helps students to recognize the processesinvolved in division, and allows them to make connec-tions between division and addition, subtraction,and multiplication

PRIOR LEARNING

Initial experiences with division in the primary grades often involve sharing objects equally

For example, students might be asked to show how 4 children could share 12 boxes of raisinsfairly Using 12 counters to represent the boxes, students might divide the counters into

4 groups while counting out, “One, two, three, four, one, two, three, four, ” until allthe “boxes” have been distributed

Students in the primary grades also apply their understanding of addition, subtraction, andmultiplication to solve division problems Consider the following problem

“Chad has 28 dog treats If he gives Rover 4 dog treats each day, for how many days will Rover get treats?”

Using addition: Students might repeatedly add 4 until they get to 28, and then count the

number of times they added 4 Students often use drawings to help them keep track of thenumber of repeated additions they make

4 + 4 + 4 + 4 + 4 + 4 + 4 = 28

Using subtraction: Students might start with 28 counters and remove them in groups of 4.

Later, students make connections to repeated subtraction (e.g., repeatedly subtracting 4 from

28 until they get to 0, and then counting the number of times 4 was subtracted)

Using multiplication: Students might use their knowledge of multiplication For example,

“Rover gets 4 treats each day Since 4×7= 28, Rover will get treats for 7 days.”

KNOWLEDGE AND SKILLS DEVELOPED IN THE JUNIOR GRADES

In the junior grades, instruction should focus on developing students’ understanding of divisionconcepts and meaningful computational strategies, rather than on having students memorizethe steps in algorithms

Development of division concepts and computational strategies should be rooted in meaningfulexperiences that allow students to model multiplicative relationships (i.e., represent a quantity as

a combination of equal groups), and encourage them to develop and apply a variety of strategies Instruction that is based on meaningful and relevant contexts helps students to achieve thecurriculum expectations related to division, listed in the following table

Curriculum Expectations Related to Division, Grades 4, 5, and 6

By the end of Grade 4, students will:

By the end of Grade 5, students will:

By the end of Grade 6, students will:

Overall Expectation

• solve problems involving theaddition, subtraction, multipli-cation, and division of single-and multidigit whole numbers,and involving the addition andsubtraction of decimal numbers

to tenths and money amounts,using a variety of strategies

Specific Expectations

• multiply to 9× 9 and divide to81÷ 9, using a variety of mentalstrategies;

• multiply whole numbers by 10,

100, and 1000, and divide wholenumbers by 10 and 100 usingmental strategies;

• divide two-digit whole numbers

by one-digit whole numbers,using a variety of tools andstudent-generated algorithms

Overall Expectation

• solve problems involving themultiplication and division ofmultidigit whole numbers, andinvolving the addition and sub-traction of decimal numbers tohundredths, using a variety ofstrategies

Specific Expectations

• divide three-digit whole numbers

by one-digit whole numbers,using concrete materials, estimation, student-generatedalgorithms, and standard algorithms;

• multiply decimal numbers by 10,

100, 1000, and 10 000, anddivide decimal numbers by 10and 100, using mental strategies;

• use estimation when solvingproblems involving the addition,subtraction, multiplication, anddivision of whole numbers, tohelp judge the reasonableness

of a solution

Overall Expectation

• solve problems involving themultiplication and division ofwhole numbers, and the addi-tion and subtraction of decimalnumbers to thousandths, using

a variety of strategies

Specific Expectations

• use a variety of mental strategies

to solve addition, subtraction,multiplication, and division prob-lems involving whole numbers;

• solve problems involving themultiplication and division ofwhole numbers (four-digit bytwo-digit), using a variety oftools and strategies;

• multiply and divide decimalnumbers to tenths by wholenumbers, using concretematerials, estimation, algo-rithms, and calculators;

• multiply and divide decimalnumbers by 10, 100, 1000, and

The following sections explain content knowledge related to division concepts in the juniorgrades, and provide instructional strategies that help students develop an understanding ofdivision Teachers can facilitate this understanding by helping students to:

• interpret division situations;

• relate multiplication and division;

• use models to represent division;

• learn basic division facts;

• consider the meaning of remainders;

• develop a variety of computational strategies;

• develop estimation strategies for division

Interpreting Division Situations

In the junior grades, students need to encounter problems that explore both partitive divisionand quotative division

In partitive division (also called distribution or sharing division), the whole amount and thenumber of groups are known, but the number of items in each group is unknown

Examples:

• Daria has 42 bite-sized granola snacks to share equally with her 6 friends How many snacksdoes each friend get?

• 168 DVDs are packaged into 8 boxes How many DVDs are there in each box?

• Zeljko’s father bought a new TV for $660 He is paying it off monthly for one year Howmuch does he pay each month?

In quotative division (also called measurement division), the whole amount and the number

of items in each group are known, but the number of groups is unknown

(Note: In this problem, students need to deal with the remainder For example, students

might conclude that more money will need to be raised one month or that an extra month

of fundraising will be needed.)

• The hardware store sells light bulbs in large boxes of 24 The last order was for 432 lightbulbs How many large boxes of light bulbs were ordered?

Students require experiences in interpreting both types of problems and in applying appropriateproblem-solving strategies It is not necessary, though, for students to identify or define theseproblem types.

Relating Multiplication and Division

Multiplication and division are inverse operations: multiplication involves combining groups

of equal size to create a whole, whereas division involves separating the whole into equalgroups In problem-solving situations, students can be asked to determine the total number

of items in the whole (multiplication), the number of items in each group (partitive division),

or the number of groups (quotative division)

Students should experience problems such as the following, which allow them to see theconnections between multiplication and division

“Samuel needs to equally distribute 168 cans of soup to 8 shelters in the city How many cans will each shelter get?”

“The cans come in cases of 8 How many cases will Samuel need in order to have 168 cans of soup?”

Although both problems seem to be division problems, students might solve the second oneusing multiplication – by recognizing that 20 cases would provide 160 cans (20× 8 =160),and that an additional case would provide another 8 cans (1× 8= 8), therefore determiningthat 21 cases would provide 168 cans With this strategy, students, in essence, decompose

168 into (20× 8) (1× 8), and then add 20+1= 21

Providing opportunities to solve related problems helps students develop an understanding

of the part-whole relationships inherent in multiplication and division situations, and enablesthem to use multiplication and division interchangeably, depending on the problem situation

Using Models to Represent Division

Models are concrete and pictorial representations of mathematical ideas It is important thatstudents have opportunities to represent division using models that they devise themselves(e.g., using counters to solve a problem involving fair sharing; drawing a diagram to represent

a quotative division situation)

Students also need to develop an understanding of conventional mathematical models fordivision, such as arrays and open arrays Because array models are also useful for representingmultiplication, they help students to recognize the relationships between the two operations.Consider the following problem

“In preparation for their concert in the gym, a class is arranging 72 chairs in rows of 12 How many rows will there be?”

a representation of 72÷12 =6 It helps students to visualize how the factors of 12 and 6 can

be combined to create a whole of 72

Teachers can also use open arrays to help students represent division situations where it isimpractical to create an array in which every square or item within the array is indicated

Consider this problem

“The organizing committee for a play day needs to organize 112 students into teams

of 8 How many teams will there be?”

Students can represent the problem using an open array

The open array may not represent how students visualize the problem (i.e., how students will

be organized into teams), and it does not provide an apparent solution to 112÷ 8 The openarray does, however, provide a tool with which students can reason their way to a solution

Students might realize that 10 teams of 8 would include 80 students but that another 32 students(the difference between 112 and 80) also need to be organized into teams of 8 By splittingthe array into sections to show that 112 can be decomposed into 80 and 32, students canre-create the problem in another way

32

?

The parts in the open array help students to determine the solution Since 32÷ 8= 4 (althoughmany students will likely think “4× 8=32”), students can determine that the number of teamswill be 10 + 4, or 14.

Initially, students use mathematical models, such as open arrays, to represent problem situationsand their own mathematical thinking With experience, students can also learn to use models

as powerful tools to think with (Fosnot & Dolk, 2001) Appendix 4–1: Using MathematicalModels to Represent Division provides guidance to teachers on how they can help studentsuse models as representations of mathematical situations, as representations of mathematicalthinking, and as tools for learning

Learning Basic Division Facts

A knowledge of basic division facts supports students in understanding division concepts and

in carrying out mental computations and paper-and-pencil calculations Because multiplicationand division are related operations, students often use multiplication facts to answer corre-sponding division facts (e.g., 4×6 = 24, so 24 ÷ 4 = 6)

The use of models and thinking strategies helps students to develop knowledge of basic facts

in a meaningful way Chapter 10 in A Guide to Effective Instruction in Mathematics, Kindergarten

to Grade 6, 2006 (Volume 5) provides practical ideas on ways to help students learn basic

division facts

Considering the Meaning of Remainders

The following problem was administered to a stratified sample of 45 000 students nationwide

on a National Assessment of Educational Progress secondary mathematics exam

“An army bus holds 36 soldiers If 1128 soldiers are being bussed to their training site, how many buses are needed?”

Seventy percent of the students completed the division computation correctly However, inresponse to the question “How many buses are needed?”, 29 percent of students answered

“31 remainder 12”; 18 percent answered “31”; 23 percent answered “32”, the correct response(Schoenfeld, 1987)

The preceding example illustrates the impact that a mathematics program focusing on learningalgorithms can have on students’ ability to interpret mathematical problems and their solutions.The example also highlights the importance of considering the meaning of remainders indivision situations

In a problem-solving approach to teaching and learning mathematics, students must considerthe meaning of remainders within the context of the problem Consider this problem

“There are 11 players on a soccer team 139 students signed up for an intramural soccer league How many teams will there be?”

In solving the problem, students discover that there are 12 teams, and 7 extra players Thesolution requires students to consider what can be done with the 7 additional players Somestudents might distribute these players to 7 teams, whereas others might suggest smaller teams

The following problem, which involves the same numbers as in the preceding situation butwith a different context, requires students to think differently about the remainder

“11 classmates purchased a painting for their teacher, who was moving to a new school.

If the painting cost $139, how much did each classmate contribute for the gift?”

In this problem, students discover that each classmate contributes $12 but that the classmatesare still short $7 Students would have to come up with a fair way to account for the shortfall

Students can deal with remainders in division problems in several ways:

• The remainder can be discarded

“Alexandrea cuts 1 m of string into 30 cm pieces How many pieces can she make?” (3 pieces,and the remaining 10 cm is discarded)

• The remainder can be partitioned into fractional pieces and distributed equally

“If 4 people share 5 loaves of bread, how much does each person get?” (1 and 1/4 loaves)

• The remainder can remain a quantity

“Six children share 125 beads How many beads will each child get?” (20 beads, with 5 beadsleft over)

• The remainder can force the answer to the next highest whole number

“Josiah needs to package 80 cans of soup in boxes Each box holds 12 cans How manyboxes does he need?” (7 boxes, but one box will not be full)

• The quotient can be rounded to the nearest whole number for an approximate answer

“Tara and her two brothers were given $25 to spend on dinner About how much moneydoes each person have to spend?” (about $8)

Presenting division problems in a variety of meaningful contexts encourages students to thinkabout remainders and determine appropriate strategies for dealing with them

Developing a Variety of Computational Strategies

Developing effective computational strategies for solving division problems is a goal ofinstruction in the junior grades However, a premature introduction to a standard divisionalgorithm does little to promote student understanding of the operation or of the meaningbehind computational procedures In classrooms where rote memorization of algorithmic steps

is emphasized, student often make computational errors without understanding why they

are doing so The following example illustrates an error made by a student who does notunderstand the division processes represented in an algorithm:

The student constructs the algorithm in his own mind as, “Come as close to the number asyou can, then subtract.” Recalling multiplication facts, he knows that 9×8 is 72 (a product that

is very close to 71) and subsequently subtracts incorrectly

EARLY STRATEGIES FOR PARTITIVE DIVISION PROBLEMS

Students are able to solve division problems long before they are taught procedures for doing

so When students are presented with problems in meaningful contexts, they rely on strategiesthat they already understand to work towards a solution In the primary grades, students oftensolve partitive division problems by dealing out or distributing concrete objects one by one.When students use this strategy to divide larger numbers, they realize that dealing out objectsone by one can be cumbersome, and that it is difficult to represent large numbers usingconcrete materials

In the junior grades, students learn to employ more sophisticated methods of fair sharing

as they develop a greater understanding of ways in which numbers can be decomposed

“Jamie’s grandmother brought home 128 shells from her beach vacation She wants to divide the shells equally among her 4 grandchildren How many shells will each grandchild receive?”

To solve this problem, students might first think of 128 as 100 + 28 They realize that 100 isfour 25’s and begin by allocating 25 to each of 4 groups Students might then distribute theremaining 28 by first allocating 5 and then 2 to each group, or they might recognize that 28

is a multiple of 4 (4×7=28) and allocate 7 to each group After distributing 128 equally to 4groups, students solve the problem by recognizing that each grandchild will receive 32 shells.The following illustration shows how students might represent their strategy

128

25 5 2 32

25 5 2 32

25 5 2 32

25 5 2 32

81

9 71672

R 7

1697

The strategy of decomposing the dividend into parts (e.g., decomposing 128 into 100 + 28)

and then dividing each part by the divisor is an application of the distributive property.

According to the distributive property, division expressions, such as 128 ÷ 4, can be split into

smaller parts, for example, (100 ÷ 4) + (28÷ 4) The sum of the partial quotients (25+ 7) provides

the answer to the division expression

EARLY STRATEGIES FOR QUOTATIVE DIVISION PROBLEMS

Division is often referred to as “repeated subtraction” (e.g., 24 ÷ 6 is the same as 24 – 6 – 6 – 6 – 6)

Although this interpretation of division is correct, students in the early stages of learning

division strategies often use repeated addition to solve quotative problems For many students, it

makes more sense to start at zero and add up to the dividend

“144 baseballs are placed in trays for storage Each tray holds 24 balls How many trays are needed?”

To solve this problem, students might repeatedly add 24 until they get to

144, and then count the number of times they added 24 to determinethe number of groups of 24, as shown at right

Students might also use repeated subtraction in a similar way Beginningwith 144, they continually subtract 24 until they get to 0, and thencount the number of times they subtracted 24

Students demonstrate a growing understanding of multiplicative ships when they realize that they can add or subtract “chunks” (groups

relation-of groups), rather than adding or subtracting one group at a time

“The library just received 56 new books The librarian wants to create take-home book packs with 4 books in each pack How many packs can he make?”

Two methods, both involving “chunking”, are illustrated in the following strategies In thefirst example (on the left), a familiar fact, 5× 4, is used to determine that 5 packs can be createdwith 20 books, and therefore 10 packs can be created with 40 books Another fact, 2×4, is used

to determine that there are 4 packs for the remaining 16 books In the second example (on theright), the same multiplication facts help to determine quantities that can be subtracted from 56

24+ 2448+ 2472+ 2496+ 24120+ 24144

56– 2036– 2016– 88– 80

It is important to note that both methods make use of the distributive property In the firstexample, 56 is decomposed into (5× 4) + (5× 4) + (2 × 4) + (2 × 4) In the second example, thenumber of 4’s is found by decomposing 56÷ 4 into (20÷ 4) + (20 ÷ 4) + (8 ÷ 4) + (8 ÷ 4) Providingopportunities for students to explore informal division strategies (which are often based on thedistributive property) prepares students for understanding more formal methods and algorithms.

DEVELOPING AN UNDERSTANDING OF THE DISTRIBUTIVE PROPERTY

The distributive property is the basis for a variety of division strategies, including the standardalgorithm An understanding of how the property can be applied in division allows students

to develop flexible and meaningful strategies, and helps bring meaning to the steps involved

Students can use an open array to model the strategy

There is significant flexibility in using the distributive property to solve division problems.For example, the preceding division expression could have been calculated by decomposing

195 into 75 and 120, then dividing 75 ÷ 15 and 120 ÷ 15, and then adding the partial quotients(5 + 8) However, strategies that use the distributive property are most effective when divisionexpressions can be broken into friendly numbers and are easy to compute For example, 150 ÷ 15and 45 ÷ 15 are generally easier to compute mentally than 75 ÷ 15 and 120 ÷ 15 are

Students learn that facts involving 10× and 100 × are helpful when using the distributiveproperty To solve 889 ÷ 24, for example, students might take a “stepped” approach todecomposing 889 into groups of 24

Students calculate that 37 groups of 24 is 888, and therefore the solution is 889 ÷ 24=37 R1

The strategy can be illustrated by using an open array

When division involves large numbers, informal strategies make it difficult for students tokeep track of numerical operations In these situations, algorithms become useful to helpstudents record and keep track of the multiple steps and operations in division

DEVELOPING AN UNDERSTANDING OF FLEXIBLE DIVISION ALGORITHMS

Flexible division algorithms, like the standard algorithm, are based on the distributive property

With flexible algorithms, however, students use known multiplication facts to decompose thedividend into friendly “pieces”, and repeatedly subtract those parts from the whole until nomultiples of the divisor are left Students keep track of the pieces as they are “removed”, which

is illustrated in the two examples below

17

10

10

222

5562 ÷ 26 = 213 R24

5562– 26002962– 2600362– 260102– 5250– 2624

387 ÷ 17 = 22 R13

A student who is using a flexible algorithm to solve the first example, 387 ÷ 17, might reason

as follows:

“I need to divide 387 into groups of 17 How many groups can I make? I know I can get

at least 10 groups That’s 170, and if I remove that, I have 217 left Another 10 groups would leave me with 47 I can get 2 groups from that, so I can take off another 34

That leaves me with 13, which isn’t enough for another group So altogether, I made

As students become more comfortable multiplying and dividing by multiples of 10, they learn

to compute using fewer partial quotients in the algorithm, as illustrated below:

DEVELOPING AN UNDERSTANDING OF THE STANDARD DIVISION ALGORITHM

Historically the algorithms (standardized steps for calculation) were created to be used for efficiency by a small group of “human calculators” when calculators were not yet invented They were not designed to support the sense making that is now expected from students.

(Teaching and Learning Mathematics in Grades 4 to 6 in Ontario, 2004, p 12)

Although the standard division algorithm provides an efficient computational method, thesteps in the algorithm can be very confusing for students if they have not had opportunities

to solve division problems using their own strategies and methods

Working with flexible division algorithms can prepare students for understanding the standardalgorithm A version of the flexible division algorithm involves stacking the quotients abovethe algorithm (rather than down the side, as demonstrated in the above example) The followingexample shows how the parts in the flexible algorithm can be connected to the recordingmethod used in the standard algorithm

387– 34047– 3413

17

20

222

387 ÷ 17 = 22 R13

Developing Estimation Strategies for Division

Students need to develop effective estimation strategies, and they also need to be aware ofwhen one strategy is more appropriate than another It is important for students to considerthe context of a problem before selecting an estimation strategy Students should also decidebeforehand how accurate their estimation needs to be Consider the following problem

“Ms Wu’s class is putting cans in boxes for the annual canned-food drive They have

188 cans and put approximately 20 cans in a box About how many boxes do they need?”

In this problem situation, it is useful to use an estimation strategy that results in enoughboxes to package all the cans (e.g., round 188 to 200 and divide by 20 to get 10 boxes)

The following table outlines different estimation strategies for division It is important to notethat the word “rounding” is used loosely – it does not refer to any set of rules or proceduresfor rounding numbers (e.g., look to the number on the right; is it greater than 5? )

the divisor accordingly

237 ÷ 11 is about 240 ÷ 12= 20

237 ÷ 11 is about 230 ÷ 10= 23Using front-end estimation

(Note that this strategy is less accurate with divisionthan with addition and subtraction.)

453 ÷ 27 is about 400 ÷ 20= 20 (actual answer is 16 R21)

Finding a range (by rounding both numbers down,then up)

565 ÷ 24 is about 500 ÷ 20= 25

565 ÷ 24 is about 600 ÷ 30= 20The quotient is between 20 and 25

125100100904– 400504– 400104– 1004– 40

4

226904– 84

10– 824– 240

A Summary of General Instructional Strategies

Students in the junior grades benefit from the following instructional strategies:

• experiencing a variety of division problems, including partitive and quotative problems;

• using concrete and pictorial models to represent mathematical situations, to representmathematical thinking, and to use as tools for new learning;

• solving division problems that serve different instructional purposes (e.g., to introducenew concepts, to learn a particular strategy, to consolidate ideas);

• providing opportunities to develop and practise mental computation and estimation strategies;

• providing opportunities to connect division to multiplication through problem solving

The Grades 4–6 Multiplication and Division module at www.eworkshop.on.ca providesadditional information on developing division concepts with students The module alsocontains a variety of learning activities and teaching resources

APPENDIX 4–1: USING MATHEMATICAL MODELS TO REPRESENT DIVISION

The Importance of Mathematical Models

Models are concrete and pictorial representations of mathematical ideas, and their use iscritical in order for students to make sense of mathematics At an early age, students usemodels such as counters to represent objects and tally marks to keep a running count

Standard mathematical models, such as number lines and arrays, have been developed overtime and are useful as “pictures” of generalized ideas In the junior grades, it is important forteachers to develop students’ understanding of a variety of models so that models can beused as tools for learning

The development in understanding a mathematical model follows a three-phase continuum:

• Using a model to represent a mathematical situation: Students use a model to represent

a mathematical problem The model provides a “picture” of the situation

• Using a model to represent student thinking: After students have discussed a mathematical

idea, the teacher presents a model that represents students’ thinking

• Using a model as a tool for new learning: Students have a strong understanding of the

model and are able to apply it in new learning situations

An understanding of mathematical models takes time to develop A teacher may be able totake his or her class through only the first or second phase of a particular model over the course

of a school year In other cases, students may quickly come to understand how the model can

be used to represent mathematical situations, and a teacher may be able to take a model tothe third phase with his or her class

USING A MODEL TO REPRESENT A MATHEMATICAL SITUATION

A well-crafted problem can lead students to use a mathematical model that the teacher wouldlike to highlight The following example illustrates how the use of an array as a model fordivision might be introduced

A teacher provides students with the following problem:

“Students in the primary division are putting on a concert, and the principal has asked our class to set up chairs in the gym for parents and guests We have 345 chairs, and the principal wants rows with 15 chairs in each row How many rows do we need to set up?”

The teacher has purposefully selected the numbers in the problem: they are friendly (easy

to work with) but large enough to prevent quick solutions They are also too large for students

to use counters or other manipulatives, and repeated subtraction or repeated addition would

be inefficient strategies To this point, the class has not been taught any formal algorithmsfor division by a two-digit number

The problem also lends itself to the use of an array Although some students attempt to solvethe problem using only numerical calculations, others use drawings to recreate the situation.One student uses grid paper, with each square representing a chair:

The student explains her strategy:

“I started drawing rows of 15 chairs I knew that 10 rows would have 150 chairs, because

would give me another 150, for a total of 300 chairs That just left 45 chairs, which is

3 rows of 15 So I know we’d have 10 rows plus 10 rows plus 3 rows, for a total of 23 rows.”

This student used an array to model the rows of chairs Although not all students used thismodel, the teacher is able to draw attention to it during the Reflecting and Connecting part

of the lesson The student, having no formal strategy for dividing by two-digit numbers, hasused an array to represent 345 chairs in rows of 15, and then has broken the array into parts

The student has also used another important idea in division – making groups of tens

Not all students used the array as a model to represent the mathematical situation, and there

is no guarantee that students who did use it can or will apply it to other division problems

It is the teacher’s role to help students generalize the use of the array as a model in otherdivision situations

USING A MODEL TO REPRESENT STUDENT THINKING

Teachers can guide students in recognizing how models can represent mathematical thinking

The following example provides an illustration

After solving problems in which the class used arrays to represent division situations, a teacherpresents the following problem:

“My neighbour is a potter and is well known for her unique coffee mugs She sells them

to kitchen stores in sets of 12, in special boxes that protect the mugs during shipping.

Yesterday, a store placed an order for 288 mugs She needs to know how many boxes she needs to ship the mugs to the store.”

The teacher encourages students to use strategies that make sense to them, and suggests thatthey use concrete materials and diagrams to help them understand and solve the problem

One student solved the problem mentally, recording the results of his mental calculations

on paper as he worked through the problem

The teacher, wanting to highlight the student’s strategy with the class, asks the student toexplain his work The student explains:

“I figured out that the problem is finding how many groups of 12 there are in 288 I started thinking about numbers I knew I knew 10 groups would be 120 mugs, and another 10 is

240 I subtracted 240 from 288 and had 48 left That’s 4 more groups of 12, so in total

288 ÷ 12 how many groups of 12?

Although the student did not use an array model to solve the problem, his teacher presents

an open array to the class to help students visualize their classmate’s thinking The solution

is represented through a series of grams

dia-“If we think about the problem

as an array, then the area of the array is 288, and the length of one side is 12 We need to find the length of the other side You solved the problem by breaking

288 into friendlier numbers:

120+ 120 + 48.”

In this case, the array is used to model a strategy in which partial quotients are determined

by using friendly numbers that are multiples of 10 The dividend has been decomposed intonumbers that are easier to work with

The teacher has provided a visual representation of a student solution that makes the strategymore accessible to other students in the class, and has built upon students’ understanding

of the array model With meaningful practice rooted in contextual problems, the open arraymodel can become a useful tool for dividing numbers

USING A MODEL AS A TOOL FOR NEW LEARNING

To help students generalize the use of an open array as a model for division, and to help themrecognize its utility as a tool for learning, teachers need to provide problems that allow students

to apply and extend the strategy of partial quotients A sample problem comes out of afundraising scenario

“23 students raised $437 for the United Way If each student brought in the same amount of money, how much did each student raise?”

The numbers in the problem were chosen to be challenging, but they also allow for variousstrategies to find a solution Many students use a strategy that involves determining partialquotients by decomposing 437 To begin, they recognize that 10× 23 =230 and draw an open

?

288 12

”I kept multiplying 23 by friendly numbers I started with 10, and got 230 I tried another

10, but that would have given me 460, which is too much So I timesed by 5, which was easy because it was half of 10 I kept going that way, trying numbers that fit Each time I tried a new number, I had to take it away from 437 to find out how much was left.

It ended up that each student raised $19.”

Another student started down a similar path but used the distributive property and subtractioninstead of addition

“I knew that 20 times 23 is 460, which was more than I needed I subtracted 437 from

460, and found the difference was 23 So that’s 1 group of 23 less, or 19 groups

1

460

23 20

student used the model to come up with a compensation strategy The second student used

the open array model as a tool for solving a division problem in a new way

When developing a model for division, it is practical to assume that not all students willcome to understand or use the model with the same degree of effectiveness Teachers shouldcontinue to develop meaningful problems that allow students to use strategies that makesense to them However, part of the teacher’s role is to use models to represent students’ideas so that these models will eventually become thinking tools for students The ability

to generalize a model and use it as a learning tool takes time (possibly years) to develop

Baroody, A J., & Ginsburg, H P (1986) The relationship between initial meaning

and mechanical knowledge of arithmetic In J Hiebert (Ed.), Conceptual and procedural

knowledge: The case of mathematics Hillsdale, NJ: Erlbaum.

Burns, M (2000) About teaching mathematics: A K–8 resource (2nd ed.) Sausalito, CA: Math

Expert Panel on Early Math in Ontario (2003) Early math strategy: The report of the Expert Panel

on Early Math in Ontario Toronto: Ontario Ministry of Education

Expert Panel on Mathematics in Grades 4 to 6 in Ontario (2004) Teaching and learning

mathematics: The report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario.

Toronto: Ontario Ministry of Education

Fosnot, C T., & Dolk, M (2001a) Young mathematicians at work: Constructing number sense,

addition, and subtraction Portsmouth, NH: Heinemann.

Fosnot, C T., & Dolk, M (2001b) Young mathematicians at work: Constructing multiplication

and division Portsmouth, NH: Heinemann.

Fosnot, C T., & Dolk, M (2001c) Young mathematicians at work: Constructing fractions,

deci-mals, and percents Portsmouth, NH: Heinemann

Fosnot, C T., Dolk, M., Cameron, A., & Hersch, S B (2004) Addition and subtraction minilessons,

Grades PreK–3 Portsmouth, NH: Heinemann.

Fosnot, C T., Dolk, M., Cameron, A., Hersch, S B., & Teig, C M (2005) Multiplication and

division minilessons, Grades 3–5 Portsmouth, NH: Heinemann.

Fuson K (2003) Developing mathematical power in number operations In J Kilpatrick,

W G Martin, & D Schifter (Eds.), A research companion to principles and standards for

school mathematics (pp 95–113) Reston, VA: National Council of Teachers of

Mathematics

Hiebert, J (1984) Children’s mathematical learning: The struggle to link form and

under-standing Elementary School Journal, 84(5), 497–513.

Kilpatrick, J., Swafford, J., & Findell, B (Eds.) (2001) Adding it up: Helping children learn

mathematics Washington, DC: National Academy Press.

Ma, L (1999) Knowing and teaching elementary mathematics Mahwah, NJ: Lawrence Erlbaum

Associates

National Council of Teachers of Mathematics (NCTM) (2001) The roles of representation in

school mathematics: 2001 Yearbook (p 19) Reston, VA: National Council of Teachers of

Mathematics

NCTM (2000) Principles and standards for school mathematics (p 67) Reston, VA: National

Council of Teachers of Mathematics

Ontario Ministry of Education (2003) A guide to effective instruction in mathematics,

Kindergarten to Grade 3 – Number sense and numeration Toronto: Author.

Ontario Ministry of Education (2004) The Individual Education Plan (IEP): A resource guide.

Toronto: Author

Ontario Ministry of Education (2005) The Ontario curriculum, Grades 1–8: Mathematics.

Toronto: Author

Ontario Ministry of Education (2006) A guide to effective instruction in mathematics, Kindergarten

to Grade 6 Toronto: Author.

Post, T., Behr, M., & Lesh, R (1988) Proportionality and the development of pre-algebra

understanding In A F Coxvord & A P Schulte (Eds.), The ideas of algebra, K–12

(pp 78–90) Reston, VA: National Council of Teachers of Mathematics

Reys, R., & Yang, D-C (1998) Relationship between computational performance and number

sense among sixth- and eighth-grade students Journal for Research in Mathematics

Education, 29(2), 225–237.

Schoenfeld, A H (1987) What’s all the fuss about metacognition? In A H Schoenfeld (Ed.),

Cognitive science and mathematics education (pp 189–215) Hillsdale, NJ: Erlbaum.

Thompson, P W (1995) Notation, convention, and quantity in elementary mathematics

In J T Sowder & B P Schappelle (Eds.), Providing a foundation of teaching mathematics in

the middle grades (pp 199–221) Albany, NY: SUNY Press.

The learning activities do not address all concepts and skills outlined in the curriculum document,nor do they address all the big ideas – one activity cannot fully address all concepts, skills, andbig ideas The learning activities demonstrate how teachers can introduce or extend mathematicalconcepts; however, students need multiple experiences with these concepts to develop astrong understanding

Each learning activity is organized as follows:

O

OVVEERRVVIIEEWW:: A brief summary of the learning activity is provided

BBIIGG IIDDEEAASS:: The big ideas that are addressed in the learning activity are identified The ways

in which the learning activity addresses these big ideas are explained

RREFLLEECCTTIINNGG AANNDD CCOONNNNEECCTTIINNGG:: This section usually includes a whole-class debriefing timethat allows students to share strategies and the teacher to emphasize mathematical concepts

HOOMMEE CCOONNNNEECCTTIIOONN:: This section is addressed to parents or guardians, and includes anactivity for students to do at home that is connected to the mathematical focus of the mainlearning activity

Reellaattiioonnsshhiippss:: By solving division problems, students explore the relationship involving a quantity,the number of groups the quantity can be divided into, and the size of each group They alsoexplore the relationships between the operations, particularly between division and repeatedsubtraction